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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The level sets of the moduli of functions of bounded characteristic


Author: Robert D. Berman
Journal: Trans. Amer. Math. Soc. 281 (1984), 725-744
MSC: Primary 30D50; Secondary 30D30
MathSciNet review: 722771
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Abstract: For $ f$ a nonconstant meromorphic function on $ \Delta = \{ \vert z\vert < 1\} $ and $ r \in (\inf \vert f\vert,\sup \vert f\vert)$, let $ \mathcal{L}(f,r) = \{ z \in \Delta :\vert f(z)\vert = r\} $. In this paper, we study the components of $ \Delta \backslash \mathcal{L}(f,r)$ along with the level sets $ \mathcal{L}(f,r)$. Our results include the following: If $ f$ is an outer function and $ \Omega $ a component of $ \Delta \backslash \mathcal{L}(f,r)$, then $ \Omega $ is a simply-connected Jordan region for which $ ({\text{fr}}\;\Omega ) \cap \{ \vert z\vert = 1\} $ has positive measure. If $ f$ and $ g$ are inner functions with $ \mathcal{L}\,(f,r) = \mathcal{L}\,(g,s)$, then $ g = \eta {f^\alpha }$, where $ \vert\eta \vert = 1$ and $ \alpha > 0$. When $ g$ is an arbitrary meromorphic function, the equality of two pairs of level sets implies that $ g = c{f^\alpha }$, where $ c \ne 0$ and $ \alpha \in ( - \infty ,\infty )$. In addition, an inner function can never share a level set of its modulus with an outer function. We also give examples to demonstrate the sharpness of the main results.


References [Enhancements On Off] (What's this?)

  • [1] F. Bagemihl and W. Seidel, Some boundary properties of analytic functions, Math. Z. 61 (1954), 186–199. MR 0065643
  • [2] E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 0231999
  • [3] Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
  • [4] O. Frostman, Potential d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions, Thèse, Lund, 1935.
  • [5] T. Hall, Sur ta mesure harmonique de certains ensembles, Ark. Mat. Astr. Fys. 25A (1937), 8pp.
  • [6] Maurice Heins, On the Lindelöf principle, Ann. of Math. (2) 61 (1955), 440–473. MR 0069275
  • [7] Maurice Heins, Selected topics in the classical theory of functions of a complex variable, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1962. MR 0162913
  • [8] A. J. Lohwater, The reflection principle and the distribution of values of functions defined in a circle, Ann. Acad. Sci. Fenn. Ser. A. I. 1956 (1956), no. no 229, 18. MR 0083040
  • [9] G. R. MacLane, Asymptotic values of holomorphic functions, Rice Univ. Studies 49 (1963), no. 1, 83. MR 0148923
  • [10] Walter J. Schneider, Approximation and harmonic measure, Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979) Academic Press, London-New York, 1980, pp. 333–349. MR 623476

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0722771-2
Keywords: Level sets, bounded characteristic, inner functions, Smirnov class
Article copyright: © Copyright 1984 American Mathematical Society